Photonics meets topology

Temporally-topological defect modes in photonic time crystals

In this paper, we investigate the properties of temporally-topological defect modes (TTDMs) (or temporally-topological interface states) in the topological photonic time crystal (PTC) systems. The PTC systems are constructed by the cascade of multiple sub-PTCs that possess temporal inversion symmetries and different topologies. The cases of two-, three-, and multiple-sub-PTC for the topological PTC system are studied. By transfer matrix method, we find that the TTDMs appear when the topological signs of the corresponding gaps in the sub-PTCs are different. The positions of TTDMs can be adjusted by changing the modulation strength of the refractive index, the time duration, and the period of the sub-PTCs. Moreover, the number of TTDMs is one less than the number of sub-PTCs. In addition, the robustness of the systems is also studied. We find that the topological PTC systems have good robustness, especially on the random configuration of the refractive index and time duration for the temporal slabs in the systems. Such research may provide a new degree of freedom for PTC applications, such as novel PTC lasers, tunable band-stop or band-suppression PTC filters, and many others, in the field of integrated photonic circuits for optical communications.

Edge State, Band Topology, and Time Boundary Effect in the Fine-Grained Categorization of Chern Insulators

We predict novel topological phases with broken time-reversal symmetry supporting the coexistence of opposite chiral edge states, which are fundamentally different from the photonic spin-Hall, valley-Hall, and higher-order topological phases. We find a fine-grained categorization of Chern insulators, their band topologies characterized by identical Chern numbers are completely different. Furthermore, we prove that different topologies cause zeros in their Bloch wave function overlaps, which imprint the band gap closing and appear at the degenerate points of topological phase transition. The Bloch wave function overlaps predict the reflection and refraction at a topological time boundary, and the overlap zeros ensure the existence of vanishing revival amplitude at critical times even though different topologies before and after the time boundary have identical Chern numbers. Our findings create new opportunities for topological metamaterials, uncover the topological feature hidden in the time boundary effect as a probe of topology, and open a venue for the exploration of the rich physics originating from the long-range couplings.

Coexistence of large-area topological pseudospin and valley states in a tri-band heterostructure system

The rapid development of topological photonics has significantly revolutionized our comprehension of electromagnetic wave manipulation in recent decades. Recent research exploiting large-area topological states inserts an additional gapless PC structure between topologically trivial and nontrivial PCs, effectively introducing the mode width degree of freedom. Nevertheless, these heterostructures mainly support only single-type waveguide states operating within a single frequency band. To address these limitations, we propose a novel, to the best of our knowledge, tri-band three-layer heterostructure system, supporting both large-area pseudospin- and valley-locked states. The system showcases tunable mode widths with different operational bandwidths. Moreover, the heterostructures exhibit inherent topological characteristics and reflection-free interfacing, which are verified in the well-designed Z-shaped channels. The proposed heterostructure system can be used to design multi-band multi-functional high-flexibility topological devices, providing great advantages for enlarging the on-chip integrated communication systems.

A second wave of topological phenomena in photonics and acoustics

Light and sound are the most ubiquitous forms of waves, associated with a variety of phenomena and physical effects such as rainbows and echoes. Light and sound, both categorized as classical waves, have lately been brought into unexpected connections with exotic topological phases of matter. We are currently witnessing the onset of a second wave of active research into this topic. The past decade has been marked by fundamental advances comprising two-dimensional quantum Hall insulators and quantum spin and valley Hall insulators, whose topological properties are characterized using linear band topology. Here, going beyond these conventional topological systems, we focus on the latest frontiers, including non-Hermitian, nonlinear and non-Abelian topology as well as topological defects, for which the characterization of the topological features goes beyond the standard band-topology language. In addition to an overview of the current state of the art, we also survey future research directions for valuable applications.

Programmed Self-Assembly of Single Colloidal Gyroids for Chiral Photonic Crystals

Gyroid structures are of extensive interest because they provide a rich platform for chiroptics as well as topological photonics. While the double-gyroid morphology as a bicontinuous structure is not uncommon in self-assembled soft materials, direct self-assembly of single-network gyroids has proven elusive. Here, an enantiomorphic pair of single-gyroid crystals comprising colloidal spheres is presented, and two distinct routes are demonstrated for programmed self-assembly of each single colloidal gyroid enantiomorph from rationally designed patchy spheres. The designer colloidal patchy spheres, which closely hew to their synthetic feasibility, are chiral, having either two staggered rectangular patches at opposite poles or four circular patches arranged in a well-defined geometry. The single colloidal gyroid, as well as its inverse structure, is shown to support a wide complete photonic bandgap in addition to exhibiting rich chiroptical properties, making them attractive chiral photonic crystals. The versatility of this single colloidal gyroid, the bottom-up routes devised here in silico, and the robustness of the design space for the chiral colloidal patchy spheres together make a strong case for single colloidal gyroids to supersede colloidal diamond, as a target for programmed self-assembly, in the quest for photonic crystals operating at optical frequencies.

Topological Metamaterials

The topological properties of an object, associated with an integer called the topological invariant, are global features that cannot change continuously but only through abrupt variations, hence granting them intrinsic robustness. Engineered metamaterials (MMs) can be tailored to support highly nontrivial topological properties of their band structure, relative to their electronic, electromagnetic, acoustic and mechanical response, representing one of the major breakthroughs in physics over the past decade. Here, we review the foundations and the latest advances of topological photonic and phononic MMs, whose nontrivial wave interactions have become of great interest to a broad range of science disciplines, such as classical and quantum chemistry. We first introduce the basic concepts, including the notion of topological charge and geometric phase. We then discuss the topology of natural electronic materials, before reviewing their photonic/phononic topological MM analogues, including 2D topological MMs with and without time-reversal symmetry, Floquet topological insulators, 3D, higher-order, non-Hermitian and nonlinear topological MMs. We also discuss the topological aspects of scattering anomalies, chemical reactions and polaritons. This work aims at connecting the recent advances of topological concepts throughout a broad range of scientific areas and it highlights opportunities offered by topological MMs for the chemistry community and beyond.

A Topological Multichannel Add-Drop Filter Based on Gyromagnetic Photonic Crystals

We theoretically proposed a topological multichannel add-drop filter (ADF) and studied its unique transmission properties. The multichannel ADF was composed of two one-way gyromagnetic photonic crystal (GPC) waveguides, a middle ordinary waveguide, and two square resonators sandwiched between them, which can be regarded as two paralleling four-port nonreciprocal filters. The two square resonators were applied with opposite external magnetic fields (EMFs) to support one-way states propagating clockwise and counterclockwise, respectively. On the basis of the fact that the resonant frequencies can be tuned by the EMFs applied to the square resonators, when the intensities of EMFs were the same, the multichannel ADF behaved as a power splitter with a 50/50 division ratio and high transmittance; otherwise, it functioned as a demultiplexer to separate two different frequencies efficiently. Such a multichannel ADF not only possesses excellent filtering performance but also has strong robustness against various defects due to its topological protection property. Moreover, each output port can be switched dynamically, and each transmission channel can operate independently with little crosstalk. Our results have the potential for developing topological photonic devices in wavelength division multiplexing systems.

Antichiral surface states in time-reversal-invariant photonic semimetals

Besides chiral edge states, the hallmark of quantum Hall insulators, antichiral edge states can exhibit unidirectional transport behavior but in topological semimetals. Although such edge states provide more flexibility for molding the flow of light, their realization usually suffers from time-reversal breaking. In this study, we propose the realization of antichiral surface states in a time-reversal-invariant manner and demonstrate our idea with a three-dimensional (3D) photonic metacrystal. Our system is a photonic semimetal possessing two asymmetrically dispersed Dirac nodal lines. Via dimension reduction, the nodal lines are rendered a pair of offset Dirac points. By introducing synthetic gauge flux, each two-dimensional (2D) subsystem with nonzero k_z is analogous to a modified Haldane model, yielding a k_z-dependent antichiral surface transport. Through microwave experiments, the bulk dispersion with asymmetric nodal lines and associated twisted ribbon surface states are demonstrated in our 3D time-reversal-invariant system. Although our idea is demonstrated in a photonic system, we propose a general approach to realize antichiral edge states in time-reversal-invariant systems. This approach can be easily extended to systems beyond photonics and may pave the way for further applications of antichiral transport.

All-optical control of topological valley transport in graphene metasurfaces

We demonstrate that the influence of Kerr effect on valley-Hall topological transport in graphene metasurfaces can be used to implement an all-optical switch. In particular, by taking advantage of the large Kerr coefficient of graphene, the index of refraction of a topologically-protected graphene metasurface can be tuned via a pump beam, which results in an optically controllable frequency shift of the photonic bands of the metasurface. This spectral variation can in turn be readily employed to control and switch the propagation of an optical signal in certain waveguide modes of the graphene metasurface. Importantly, our theoretical and computational analysis reveals that the threshold pump power needed to optically switch ON/OFF the signal is strongly dependent on the group velocity of the pump mode, especially when the device is operated in the slow-light regime. This study could open up new routes towards active photonic nanodevices whose underlying functionality stems from their topological characteristics.

Topological phase transition and robust pseudospin interface states induced by angular perturbation in 2D topological photonic crystals

In recent years the research about topological photonic structures has been a very attractive topic in nanoscience from both a basic science and a technological point of view. In this work we propose a two-dimensional topological photonic structure, composed of a trivial and a topological photonic crystals, made of dumbbell-shaped dielectric rods. The topological behavior is induced by introducing an angular perturbation in the dumbbell-shaped dielectric rods. We show that this composed structure supports pseudospin interface states at the interface between the trivial and topological crystals. Our numerical results show that a bandgap is opened in the band structure by introducing the angular perturbation in the system, lifting the double degeneracy of the double Dirac cone at the [Formula: see text] point of the Brillouin zone, despite keeping the [Formula: see text] symmetry group. A pseudospin topological behavior was observed and analyzed with emphasis on the photonic bands at the [Formula: see text] point. We have also investigated the robustness of these pseudospin interface states and, according with our numerical results, we conclude that they are robust against defects, disorder and reflection. Finally, we have shown that the two edge modes present energy flux propagating in opposite directions, which is the photonic analogue of the quantum spin Hall effect.

Different phases in non-Hermitian topological semiconductor stripe laser arrays

As a novel branch of topology, non-Hermitian topological systems have been extensively studied in theory and experiments recently. Topological parity-time (PT)-symmetric semiconductor stripe laser arrays based on the Su-Schreiffer-Heeger model are proposed. The degree of non-Hermicity can be tuned by altering the length of the cavities, and PT symmetry can be realized by patterned electrode. Three laser arrays working in different non-Hermitian phases are analyzed and fabricated. With the increasing degree of non-Hermicity, the peaks of output intensities move from the edge to the bulk. The proposed semiconductor stripe laser array can function as an active, flexible, and feasible platform to investigate and explore non-Hermitian topology for further developments in this field.

Topological Resistance-Free One-Way Transport in a Square-Hexagon Lattice Gyromagnetic Photonic Crystal

We theoretically propose and experimentally realize a new configuration of a photonic Chern topological insulator (PCTI) composed of a two-dimensional square-hexagon lattice gyromagnetic photonic crystal immersed in an external magnetic field. This PCTI possesses five distinct types of edges and all of them allowed the propagation of truly one-way edge states. We proceeded to utilize this special PCTI to design topological transmission lines of various configurations with sharp turns. Although the wave impedances of the edge states on both sides of the intersections in these transmission lines were very different, definitely no back reflection occurred and no mode-mixing problems and impedance-mismatching issues at the intersections were present, leading to topological resistance-free one-way transport in the whole transmission line network. Our results enrich the geometric and physical means and infrastructure to construct one-way transport and bring about novel platforms for developing topology-driven resistance-free photonic devices.

High-speed infrared photonic band microscope using hyperspectral Fourier image spectroscopy

In this study, we developed a photonic band microscope based on hyperspectral Fourier image spectroscopy. The developed device constructs an infrared photonic band structure from Fourier images for various wavelength obtained by hyperspectral imaging, which make it possible to speedily measure the dispersion characteristics of photonic nanostructures. By applying the developed device to typical photonic crystals and topological photonic crystals, we succeeded in obtaining band structures in good agreement with the theoretical prediction calculated by the finite element method. This device facilitates the evaluation of physical properties in various photonic nanostructures, and is expected to further promote related fields.

Highly efficient vertical coupling to a topological waveguide with defect structure

In this study, we propose a defect structure that enhances the vertical coupling efficiency of circularly polarized light incident on topological waveguides consisting of triangle nanoholes with C_6v symmetry arranged in honeycomb lattice. The defect structure was formed by removing triangle nanoholes from a certain hexagonal unit cell around the topological waveguide. As a result of comparing the coupling efficiency with and without the defect structure through three-dimensional finite-difference time-domain analysis, significant improvement in the vertical coupling efficiency was observed over the entire telecom C band (4460%@1530 nm). In addition, it was also found that the wavelength showing maximum coupling efficiency can be controlled over the entire C band by changing the arrangement of the dielectric around the defect structure.

Chiral and multiple one-way surface states on photonic gyroelectric metamaterials with small Chern number

Topological one-way surface states allow light to pass through sharp corners without reflection. In order to enhance the capability of surface routing devices, multiple one-way surface modes are usually required. Different from previously reported multiple surface modes achieved with large Chern number photonic media, we realize multiple surface waves on a continuous medium with small Chern number, i.e., |C| = 1. The new topological phase is found when the hyperbolic and double semi-ellipsoid-like cone bands are simultaneously gapped by vacuum state. We also find the degeneracy of multiple one-way surface waves in the double semi-ellipsoid-like metamaterials. The propagation direction of the waves is determined by their own ellipticities. Our results may help to construct surface state devices with multiplexing capability and higher coupling efficiency.

Recent advances in topological elastic metamaterials

Topological elastic metamaterials have emerged as a new frontier in the quest of topological phases in condensed matter physics. Their exotic topological properties open a wealth of promising engineering-oriented applications that are difficult to realize with traditional elastic metamaterials, such as robust and defect insensitive waveguiding, signal sensing, and splitting. In this review, we retrospectively examine the underlying physical concept of topologically ordered states of elastic waves, starting from the one-dimensional example based on the Su-Schrieffer-Heeger model. We then move on to two-dimensional topological metamaterials, discussing elastic analogues of quantum Hall, pseudospin-Hall, valley-Hall phases. Finally, we survey the latest developments in the field including three-dimensional elastic topological phases and higher-order topological insulators. Altogether, this paper provides a comprehensive overview of the flourishing research frontier on topological elastic metamaterials, and highlights prominent future directions in this field.

Quantum superposition demonstrated higher-order topological bound states in the continuum

Higher-order topological insulators, as newly found non-trivial materials and structures, possess topological phases beyond the conventional bulk-boundary correspondence. In previous studies, in-gap boundary states such as the corner states were regarded as conclusive evidence for the emergence of higher-order topological insulators. Here, we present an experimental observation of a photonic higher-order topological insulator with corner states embedded into the bulk spectrum, denoted as the higher-order topological bound states in the continuum. Especially, we propose and experimentally demonstrate a new way to identify topological corner states by exciting them separately from the bulk states with photonic quantum superposition states. Our results extend the topological bound states in the continuum into higher-order cases, providing an unprecedented mechanism to achieve robust and localized states in a bulk spectrum. More importantly, our experiments exhibit the advantage of using the time evolution of quantum superposition states to identify topological corner modes, which may shed light on future exploration between quantum dynamics and higher-order topological photonics.

Direct quantification of topological protection in symmetry-protected photonic edge states at telecom wavelengths

Topological on-chip photonics based on tailored photonic crystals (PhCs) that emulate quantum valley-Hall effects has recently gained widespread interest owing to its promise of robust unidirectional transport of classical and quantum information. We present a direct quantitative evaluation of topological photonic edge eigenstates and their transport properties in the telecom wavelength range using phase-resolved near-field optical microscopy. Experimentally visualizing the detailed sub-wavelength structure of these modes propagating along the interface between two topologically non-trivial mirror-symmetric lattices allows us to map their dispersion relation and differentiate between the contributions of several higher-order Bloch harmonics. Selective probing of forward- and backward-propagating modes as defined by their phase velocities enables direct quantification of topological robustness. Studying near-field propagation in controlled defects allows us to extract upper limits of topological protection in on-chip photonic systems in comparison with conventional PhC waveguides. We find that protected edge states are two orders of magnitude more robust than modes of conventional PhC waveguides. This direct experimental quantification of topological robustness comprises a crucial step toward the application of topologically protected guiding in integrated photonics, allowing for unprecedented error-free photonic quantum networks.

Mapping the design space of photonic topological states via deep learning

Topological states in photonics offer novel prospects for guiding and manipulating photons and facilitate the development of modern optical components for a variety of applications. Over the past few years, photonic topology physics has evolved and unveiled various unconventional optical properties in these topological materials, such as silicon photonic crystals. However, the design of such topological states still poses a significant challenge. Conventional optimization schemes often fail to capture their complex high dimensional design space. In this manuscript, we develop a deep learning framework to map the design space of topological states in the photonic crystals. This framework overcomes the limitations of existing deep learning implementations. Specifically, it reconciles the dimension mismatch between the input (topological properties) and output (design parameters) vector spaces and the non-uniqueness that arises from one-to-many function mappings. We use a fully connected deep neural network (DNN) architecture for the forward model and a cyclic convolutional neural network (cCNN) for the inverse model. The inverse architecture contains the pre-trained forward model in tandem, thereby reducing the prediction error significantly.

Tunable light absorption of graphene using topological interface states

A tunable light absorption of graphene using topological interface states (TISs) is presented. The monolayer graphene is embedded in the interface of asymmetric topological photonic crystals (ATPCs). A strong absorption phenomenon occurs by the excitation of TISs. It is found that the absorption spectra are intensively dependent on the chemical potential of graphene and the periodic number of the ATPCs. Furthermore, the absorption can be rapidly switched in a slight variation of chemical potential, which is modulated by the applied gate voltage on graphene. This study not only opens up a new approach for enhancing light-monolayer graphene interactions, but also provides for practical applications in high absorption optoelectronic devices. This strong absorption phenomenon is different from those in Fabry-Perot resonators, nano-cavities photonic crystal, and traditional topological photonic crystals (TPCs).

Recent advances in 2D, 3D and higher-order topological photonics

Over the past decade, topology has emerged as a major branch in broad areas of physics, from atomic lattices to condensed matter. In particular, topology has received significant attention in photonics because light waves can serve as a platform to investigate nontrivial bulk and edge physics with the aid of carefully engineered photonic crystals and metamaterials. Simultaneously, photonics provides enriched physics that arises from spin-1 vectorial electromagnetic fields. Here, we review recent progress in the growing field of topological photonics in three parts. The first part is dedicated to the basics of topological band theory and introduces various two-dimensional topological phases. The second part reviews three-dimensional topological phases and numerous approaches to achieve them in photonics. Last, we present recently emerging fields in topological photonics that have not yet been reviewed. This part includes topological degeneracies in nonzero dimensions, unidirectional Maxwellian spin waves, higher-order photonic topological phases, and stacking of photonic crystals to attain layer pseudospin. In addition to the various approaches for realizing photonic topological phases, we also discuss the interaction between light and topological matter and the efforts towards practical applications of topological photonics.

Topological valley plasmon transport in bilayer graphene metasurfaces for sensing applications

Topologically protected plasmonic modes located inside topological bandgaps are attracting increasing attention, chiefly due to their robustness against disorder-induced backscattering. Here, we introduce a bilayer graphene metasurface that possesses plasmonic topological valley interface modes when the mirror symmetry of the metasurface is broken by horizontally shifting the lattice of holes of the top layer of the two freestanding graphene layers in opposite directions. In this configuration, light propagation along the domain-wall interface of the bilayer graphene metasurface shows unidirectional features. Moreover, we have designed a molecular sensor based on the topological properties of this metasurface using the fact that the Fermi energy of graphene varies upon chemical doping. This effect induces strong variation of the transmission of the topological guided modes, which can be employed as the underlying working principle of gas sensing devices. Our work opens up new ways of developing robust integrated plasmonic devices for molecular sensing.

Robust high-Q filter with complete transmission by conjugated topological photonic crystals

High quality factor (High-Q) and transmission optical devices are required for various applications in the fields of physics and engineering. Critical for these applications is the realization of a structure with high-Q, complete transmission and small volume. A robust high-Q filter with complete transmission by conjugated topological photonic crystals (CTPC) is presented. The study shows that an ultra-high-Q of more than 108 with complete transmission is obtained by the CTPC with 2 μm long due to the excitation of conjugated topological edge-states (CTES). It is also found that even though the quality factor of resonances increases as the periodic number of multilayers increases, these resonances are still complete transmission. A novel concept of CTES is first proposed in this study and investigated the effect of its topological phenomenon on high quality factor via CTPC. We theoretically realize the robust high-Q and complete transmission in the CTPC, which is different from those in periodic, quasi-periodic, Fabry-Perot photonic crystals and traditional topological photonic crystals (TPC).

First-principle calculation of Chern number in gyrotropic photonic crystals

As an important figure of merit for characterizing the quantized collective behaviors of the wavefunction, Chern number is the topological invariant of quantum Hall insulators. Chern number also identifies the topological properties of the photonic topological insulators (PTIs), thus it is of crucial importance in PTI design. In this paper, we develop a first principle computatioal method for the Chern number of 2D gyrotropic photonic crystals (PCs), starting from the Maxwell's equations. Firstly, we solve the Hermitian generalized eigenvalue equation reformulated from the Maxwell's equations by using the full-wave finite-difference frequency-domain (FDFD) method. Then the Chern number is obtained by calculating the integral of Berry curvature over the first Brillouin zone. Numerical examples of both transverse-electric (TE) and transverse-magnetic (TM) modes are demonstrated, where convergent Chern numbers can be obtained using rather coarse grids, thus validating the efficiency and accuracy of the proposed method.

Dimensional hierarchy of higher-order topology in three-dimensional sonic crystals

Wave trapping and manipulation are at the heart of modern integrated photonics and acoustics. Grand challenges emerge on increasing the integration density and reducing the wave leakage/noises due to fabrication imperfections, especially for waveguides and cavities at subwavelength scales. The rising of robust wave dynamics based on topological mechanisms offers possible solutions. Ideally, in a three-dimensional (3D) topological integrated chip, there are coexisting robust two-dimensional (2D) interfaces, one-dimensional (1D) waveguides and zero-dimensional (0D) cavities. Here, we report the experimental discovery of such a dimensional hierarchy of the topologically-protected 2D surface states, 1D hinge states and 0D corner states in a single 3D system. Such an unprecedented phenomenon is triggered by the higher-order topology in simple-cubic sonic crystals and protected by the space group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${P}_{m\bar{3}m}$$\end{document}Pm3 ¯m. Our study opens up a new regime for multidimensional wave trapping and manipulation at subwavelength scales, which may inspire future technology for integrated acoustics and photonics.

Here, the authors report the experimental discovery of such a dimensional hierarchy of the topologically-protected 2D surface states, 1D hinge states and 0D corner states in a single 3D acoustic system by using higher-order topological sonic crystals.

Visualization of Higher-Order Topological Insulating Phases in Two-Dimensional Dielectric Photonic Crystals

The studies of topological phases of matter have been developed from condensed matter physics to photonic systems, resulting in fascinating designs of robust photonic devices. Recently, higher-order topological insulators have been investigated as a novel topological phase of matter beyond the conventional bulk-boundary correspondence. Previous studies of higher-order topological insulators have been mainly focused on the topological multipole systems with negative coupling between lattice sites. Here we experimentally demonstrate that second-order topological insulating phases without negative coupling can be realized in two-dimensional dielectric photonic crystals. We visualize both one-dimensional topological edge states and zero-dimensional topological corner states by using the near-field scanning technique. Our findings open new research frontiers for photonic topological phases and provide a new mechanism for light manipulating in a hierarchical way.

Band structure and topological phase transition of photonic time crystals

We investigate the band structure and topological phase transition of photonic time crystals (PTC)-systems in which the physical parameter varies periodically in time. We find that the topological phase transition of the PTC system can be revealed by the wave vector gap (k-gap) size, which was induced by the temporal refraction and reflection. Interestingly, a temporal zero-averaged refractive index k-gap is obtained when the PTC system includes a dispersive medium. This special k-gap is invariant with modulation time scaling at a given modulation frequency.

Feature issue introduction: Topological Photonics and Materials

Photonic crystals have become a very common and powerful concept in optics since its introduction in the 1980s by Eli Yablonovitch and Sajeev John. It is in fact a concept borrowed from condensed matter physics. The discussion of photonic bands and bandgaps allows us to manipulate light on an optical chip, along a photonic crystal fiber and even in the quantum optics regime. Now, we are witnessing another round of concept translation again from condensed matter physics to optics about topology. Describing photonic bands by using their topology in the reciprocal space gives us a new tool to understand wave propagation and to design optical components. Topology is also an important aspect in light-matter interaction in the field of metamaterials and 2D materials.